all instruction was conducted ideally.
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to their article, which may be described as a collection of problems,
Gerver, Konstantinov, and Kushnerenko (1965) write:
The problems presented here constitute a course in calculus. The
collection contains the necessary definitions for independently solving
all problems. By going over the material in this way, students master
the techniques of mathematical thinking step by step. To master such
techniques on a serious, professional level is the main aim of the
course. (p. 41)
Obviously, a course constructed in this way implies a teaching
process organized in a special manner. Davidovich, Pushkar’, and
Chekanov (2008), teachers at Moscow’s school No. 57 who use this
approach to teaching, preface their collection of “sheets” by explaining
that five or six teachers must be present in the classroom at the same
time. The “sheets” are handed out to the students (sometimes this is
preceded by some brief explanation) and the students then solve them
(at home or in class) and hand in their work to the teacher:
The teacher can also discuss other ways of solving the same problems,
go back to problems from older sheets that are connected with a new
topic, formulate new definitions, and pose new problems (and receive
their solutions from the students). One of the most important goals
in all this is to fill in the “empty spaces” between problems, to create
a holistic picture of the area being studied. (pp. 8–9)
Naturally, not all courses in all schools are structured in this
manner. In the overwhelming majority of cases, lessons are outwardly
quite traditional: there is one teacher who cannot listen to many
responses simultaneously. Nonetheless, structuring a lesson as a system
of problem-solving sessions, during the course of which students
acquire the desired knowledge, is quite typical. R. Gordin, a teacher
at the same school No. 57 who teaches geometry in the traditional
manner (see, for example, Gordin, 2006), emphasized in an interview
with us (2005) that problem solving usually arises in the course of
class discussions, when students gradually improve and supplement
one another’s suggestions. The ability to structure a lesson in a
corresponding manner, both in terms of selecting problems and in
terms of organizing the discussion, is therefore quite important.
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The most varied forms of working with problems are used: students
are assigned problems for long-term periods and, conversely, they are
given question-problems that require a quick response — make a pre-
diction, formulate a hypothesis, or find a mistake; different solutions to
the same problem are examined in class; oral and written problems are
combined; and so on (Karp, 1992, 2010b). Once again, this does not
mean that there can be no in-class lectures, explanations by teachers,
or simply workshops during which students solve relatively routine
(even if sufficiently technically difficult) exercises. All of this is also
possible: a lecture that contextualizes what has been learned, analyzes
what has been achieved, and poses new problems can sometimes be
no less useful than the problem-solving sessions described above, nor
can certain skills be formed without practice. There are also examples
of an approach to teaching that outwardly resembles the traditional
lecture–seminar system (Dynkin, 1967). What is important is that the
spirit of research and the independent search for truth not be replaced
by craftsmanship and the execution of commands and algorithms,
however difficult they might be.
Below, we will discuss certain sections of the course taught in
mathematics schools, including what would appear to be traditional
college topics. It must be emphasized, therefore, that the “assimilation
of new ideas and concepts,” with which Shvartsburd connected the
very notion of advanced preparation in mathematics in the passage
quoted above, by no means implied “covering” college courses as
quickly as possible: it was never anyone’s goal to report cheerfully
that students had already gone through, say, ordinary differential
equations, or even partial differential equations, while they were still
in school. The point was understood to be precisely the opposite:
to examine what was being studied more attentively (and often for
longer periods of time) than this was done in college. The aim was not
only and even not mainly to learn a particular topic, but to develop
“the techniques of mathematical thinking,” as Gerver, Konstantinov,
and Kushnerenko stated in the quote above. It is another matter that
developing such techniques is impossible without a serious command
of specific concrete mathematical material. What such material might
consist of is the topic to which we will now turn.
